C₇×#10531 #13187747

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10530
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#10531

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

386 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(8,9)(10,20)(11,15)(12,18)(13,17)(14,19)(16,22)(21,23)(24,25)(26,33)(27,29)(28,32)(30,36)(31,35)(34,37)(38,39);(8,10)(9,15)(11,13)(12,22)(14,21)(16,19)(17,23)(18,20)(25,29)(26,34)(27,35)(30,37)(31,32)(33,38);(8,11,12,21,14,22,13,10)(9,16,17,23,19,15,18,20)(24,26,35,39,28,34,27,36)(25,30,29,37,32,38,31,33)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3519);